The Two-Dimensional Hubbard Model
6.1 The Hubbard Hamiltonian
The starting point of each DMFT and DCA calculation is the definition of the model Hamiltonian ˆHlatt for the underlying infinitely large two-dimensional lattice. In the case of the Hubbard model the system is described by
Hˆlatt=X
~k,σ
ε~kd˜†~k,σd˜~k,σ+UX
i
ni,↑ni,↓−µX
i
(ni,↑+ni,↓)
=X
~k
ε~kN˜~k+U 2
X
i
Ni(Ni−1)−µX
i
Ni. (6.1.1)
with the typical Hubbard interaction on each site iof the real-space lattice with strength U and a chemical potential µ to control the filling of the lattice. The particle number operators are defined as
Ni =ni,↑+ni,↓ =d†i,↑di,↑+d†i,↓di,↓, (6.1.2) N˜~k = ˜d~†k,↑d˜~k,↑+ ˜d~†k,↓d˜~k,↓. (6.1.3) The first term of ˆHlatt represents the kinetic energy written in momentum space. To make the notation clearer, we denote all creation and annihilation operators acting in momen-tum space with a tilde. The kinetic energy of the single-particle states in momenmomen-tum space is given by the dispersion relation
ε~k =−2t(coskx+ cosky)−4t0coskxcosky, (6.1.4) with nearest neighbour hopping t = 1, next-nearest neighbour hopping t0 =−0.15t, and two-dimensional momentum~k = (kx, ky).
As described in the previous section, in the context of DMFT an auxiliary impurity problem is created by taking a single site out of the infinitely large lattice. On this site, called impurity, the same interaction as on each site of the lattice is present. Thus, we define the interaction Hamiltonian of our impurity model as
Hˆint= U
2N(N −1), (6.1.5)
and the single-particle on-site energy as
Hˆsp=−µN, (6.1.6)
both acting on the impurity site. The kinetic term vanishes since the dispersion relation averaged over the whole Brillouin zone is zero. Furthermore, the impurity is coupled via single-particle hopping to a non-interacting environment called bath. The coupling between impurity and environment is completely described by the hybridisation ∆(iωn), which has to be determined self-consistently within DMRG. A discretisation of the bath
into a finite number of Lb bath sites allows to write down the coupling and the bath Hamiltonian of the impurity problem as
Hˆhyb =X
l,σ
Vl,σd†σcl,σ+ h.c., (6.1.7)
Hˆbath =X
l,σ
l,σc†l,σcl,σ, (6.1.8)
where c†l,σ and cl,σ denote the creation and annihilation operators of an electron on bath site l. The on-site energies of the bath sites l,σ and the single-particle hopping elements Vl,σ are obtained by a fit of the hybridisation ∆(iωn). Combining all terms, we obtain the complete Hamiltonian describing the impurity problem
Hˆimp = ˆHint+ ˆHsp+ ˆHhyb+ ˆHbath. (6.1.9) All DMFT results in this chapter are obtained with ˆHimp.
When using multi-site DCA the resulting Hamiltonian is more complex. It is very im-portant for entanglement properties and implementational details whether the impurity Hamiltonian is represented in real or momentum space. We will first show the momentum-space representation and afterwards transform the Hamiltonian to real momentum-space.
As discussed in the previous section, in the case of multi-site DCA each impurity site describes a single patch of the Brillouin zone. Thus, the number of patches is equivalent to the number of impurity sites Nimp. As for DMFT, in real-space the same interaction as in the original lattice model is present on each site
Hˆintreal =NXimp
i=1
U
2Ni(Ni−1). (6.1.10)
The interaction Hamiltonian ˆHintrealhas to be Fourier transformed, as defined in Eq. (5.4.12), Eq. (5.4.13) and Tab. 5.2, to obtain the impurity Hamiltonian in momentum space Hˆintmom = U
2
X
σ,σ0 Nimp
X
r,s,m,n=1
δ(K~r−K~s+K~m−K~n) ˜d†r,σd˜s,σd˜†m,σ0d˜n,σ0 −U 2
X
i
N˜i. (6.1.11)
The termδ(K~r−K~s+K~m−K~n) together with the representative momentum vectors K~ of the DCA patches assures that only two-particle interactions are allowed that conserve the total two-dimensional momentum. In general, those terms create a lot of entangle-ment between the impurity sites and between their corresponding bath sites, especially for higher DCA approximations.
The kinetic term in the momentum-space representation corresponds to an on-site en-ergy and is obtained by averaging over the the dispersion relation inside of each patch
˜ti = P~k∈P~
Ki ε~k. As discussed in section 5.4, this guarantees the correct high frequency
behaviour of the Matsubara Green’s function. Combining these terms with the on-site energy given by µ, we obtain
Hˆspmom=−
N˜imp
X
i=1
(µ−˜ti) ˜Ni. (6.1.12)
Because each impurity site corresponds to an momentum K~ and because the model ex-plicitly conserves the momentum between the patches, no single-particle hopping exists between the impurity sites in momentum space. This is reflected by the fact that the hy-bridisation, the self-energy, and the Green’s function are all diagonal matrices. Therefore, each impurity site has an independent hybridisation, which can be fitted efficiently and in parallel with only a few parameters corresponding to a small number of bath sitesLb,i. This results in significant runtime improvements for large systems. In general, each bath can consist of a different number of bath sites, but in most cases they are all set to the same size Lb,i =Lb for simplicity.
Since each impurity site can only couple to its own bath, we can label each bath site by an index i for the impurity it belongs to and an index l numbering the Lb sites of this bath. Thus, we can write the bath and the coupling Hamiltonian in momentum space as Hˆhybmom= X
l,i,σ
Vl,i,σd˜†i,σcl,i,σ+ h.c., (6.1.13)
Hˆbathmom= X
l,i,σ
l,i,σc†l,i,σcl,i,σ. (6.1.14)
The bath sites are not denoted with a tilde because they are neither located in momentum nor in real space. The complete Hamiltonian is obtained by summing all terms
Hˆimpmom= ˆHintmom+ ˆHspmom+ ˆHhybmom+ ˆHbathmom. (6.1.15) If we refer to momentum-space DCA, all results are obtained with ˆHimpmom.
Obviously, the whole Hamiltonian can also be represented in real space by applying the inverse Fourier transform on the impurity sites. The advantage of the real-space repre-sentation is that the interaction on the impurity sites is much simpler and completely local
Hˆintreal =NXimp
i=1
U
2Ni(Ni−1), (6.1.16)
because it only consists of the typical Hubbard interaction on each site. Since DCA restores some of the momentum dependence of the original lattice model, single-particle hopping between the impurity sites exists in real space
Hˆspreal =−
Nimp
X
i=1
µNi+X
σ Nimp
X
i,j=1 i6=j
ti,jc†i,σcj,σ, (6.1.17)
with hopping strength ti,j = 1
Nimp Nimp
X
n=1
t˜ne−(R~i−R~j)K~n. (6.1.18)
Mathematically, the finite hopping elements originate from the difference of the on-site energies of the impurities in momentum space. If all on-site energies were the same,
˜tn = ˜t, Eq. (6.1.18) would result in a Kronecker delta and the single-particle hopping would vanishti,j =δi,jt.
While the Hamiltonian of the non-interacting bath stays unchanged under the transfor-mation of the impurity sites, the form of the coupling Hamiltonian changes significantly
Hˆhybreal = X
l,i,j,σ
V˜l,i,j,σd†j,σcl,i,σ+ h.c., (6.1.19)
Hˆbathreal = X
l,i,σ
l,i,σc†l,i,σcl,i,σ. (6.1.20)
Due to the Fourier transform, each bath site is now coupling with each impurity site with coupling strength
V˜l,i,j,σ = 1
qNimp
Vl,i,σe−R~jK~i. (6.1.21)
This not only creates a complicated impurity system with a lot of single-particle hopping terms between many sites but also results in hybridisations, Green’s functions, and self-energies having non-zero off-diagonal elements. Thus, fitting the hybridisation turns into a highly complicated and not well understood mathematical problem. Fortunately, it is always possible to fit the diagonal hybridisation in momentum space and transform the complete Hamiltonian into real space.
After we defined all necessary terms, the whole real-space Hamiltonian can be written as Hˆimpreal = ˆHintreal+ ˆHspreal+ ˆHhybreal+ ˆHbathreal. (6.1.22) It is not known beforehand and has to be determined for each model heuristically whether the real-space or momentum-space representation of the DCA Hamiltonian is better suited for a specific problem in terms of entanglement and convergence properties.
However, we now focus on the symmetries present in the Hubbard model regardless of its representation. We want to emphasise that the DMFT Hamiltonian can be seen as a special case of ˆHimpreal with only one impurity Nimp = 1 and thus will not be mentioned specifically. Furthermore, the real-space and momentum-space representation of the Hub-bard model are connected by a unitary Fourier transformation, which does not change the symmetry properties of the model. Therefore, for simplicity, we will only refer to ˆHimpmom in the following discussion.
• The full Hamiltonian is clearly conserving the total particle number of the system, which is described by an U(1)-symmetry with quantum number N since only pairs of creation and annihilation operators occur in ˆHimpmom.
• The length of the total spin||S~||2 is also conserved because all pairs of annihilation and creation operators in the Hamiltonian act on electrons with the same spin. This corresponds to anSU(2)-symmetry with quantum numberSvia||S~||2 =S(S+1/2).
• The Hubbard model also conserves the total momentum of the whole system. As described in section 5.4, each impurity site and also its corresponding bath sites in momentum-space belong to a patch in the first Brillouin zone. Each of those patches is associated with a representative momentum vector K~i. Adding or removing a particle from an impurity site or a corresponding bath site will add or remove the associated momentum vector K~i from the total momentum of the system. The conservation of the total momentum in the interaction term ˆHintmomis ensured by the Kronecker delta. Since each bath site only couples via single-particle hopping to a single impurity, it is clear that the whole Hamiltonian ˆHimpmom conserves the total momentum.
Due to the choice of the representative vectors in Tab. 5.2, we can describe the total momentum occurring in our multi-site DCA calculations by
K~ = (kx, ky) = π 2n,π
2m
, (6.1.23)
with n, m ∈ N0 denoting the necessary two quantum numbers. Because the first Brillouin zone {(kx, ky)|kx, ky ∈ [−π, π]} is 2π-periodic, each momentum vector with quantum numbers n >3 orm >3 corresponds to a vector inside the Brillouin zone described by quantum numbers n, m ∈ {0,1,2,3}. Thus, the momentum conservation is described by aZ4×Z4-symmetry group.
As an example we will consider a two-site DCA system with k particles on the impurity site and bath sites corresponding to the first patch and l particles on the impurity site and bath sites belonging to the second patch. Then, the total momentum of the system is given by
K~ = (kx, ky) = k·(0,0) +l·(π, π) = (lπ, lπ) =
(0,0) for l even
(π, π) forl odd , (6.1.24) because of the translational properties of the Brillouin zone. The corresponding quantum numbers are
(n, m) = k·(0,0) +l·(2,2) = (2l,2l) =
(0,0) for l even
(2,2) for l odd , (6.1.25) due to the group properties of Z4×Z4. Thus, the total momentum of the system can be determined by the quantum numbers. The same is valid, even if more com-plicated, for four-site and eight-site DCA.
The number and the choice of the patches, or more specific the choice of the rep-resentative vectors, determines which symmetry group can be used to describe the
momentum quantum numbers. In the case of two-site DCA and four-site DCA, all momentum vectors can also be described by K~ = (kx, ky) = (nπ, mπ) with n, m ∈ N0. Then, by the same argumentation as above, the quantum numbers belong to the symmetry group Z2 ×Z2. That both symmetry groups can be used in those cases is not surprising because Z2 is a subgroup of Z4. For two-site and four-site DCA not all allowed values for n and m are used when using the Z4×Z4
symmetry group, but only the even values, which directly correspond to the sub-groupZ2×Z2. Similarly, the quantum numbers for eight-site and 16-site DCA with the representative vectors chosen in Tab. 5.2 can only be described by the Z4×Z4
symmetry group or by groups that include this symmetry.
In our calculations we limit the ground state search to symmetry sectors with total momentum K~ = (0,0) since up to now, global ground states were always located in this sector for all parameters choices. This is reasonable since states with a higher total momentum have higher kinetic energies and are, in general, not the ground states of impurity systems.
Our experience with the Hubbard model implies that, in general, the real-space represen-tation of the model generates less entanglement than the momentum-space represenrepresen-tation.
This results in significantly different runtimes and convergence properties in favour of a real-space description. This difference is even increasing for DCA approximations with a higher number of impurity sites.
Unfortunately, the momentum quantum numbers can not be implemented in real space.
A necessary condition for implementing a quantum number is that the change of all quantum numbers of the system has to be well-defined for any application of creation or annihilation operators. This must be clear independently of the state of the rest of the system. In momentum space, adding a particle to an impurity site will increase the total momentum by the associated momentum vector regardless of the rest of the system. A creation operator d†j in real space corresponds, via the Fourier transform, to a sum of creation operators acting on all impurity sites in momentum space ˜d†k. Without having information about the occupation of all other impurity sites in real space, and thus about the occupation in momentum space, it is not clear how the creation operatord†j is chang-ing the quantum numbers of the total momentum.
This is problematic because we observed significant convergence problems for higher order DCA calculations when not using momentum quantum numbers. Our DMRG calculation are always initialised with a random state. If a symmetry is not explicitly implemented, it may happen that the starting state of DMRG is located in a part of the Hilbert space that is orthogonal to the ground state. But even if a part of the randomly generated starting state has a non-orthogonal component with respect to the correct ground state, it can happen that DMRG is discarding these parts first few iterations. Both situations prevent DMRG from ever reaching the true ground state. To avoid this, we strongly rec-ommend to use the momentum-space representation of the Hubbard model and implement momentum quantum numbers.